Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles

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DOI:10.1051/m2an/2011008 www.esaim-m2an.org

SOME ENERGY CONSERVATIVE SCHEMES FOR VIBRO-IMPACTS OF A BEAM ON RIGID OBSTACLES

^∗

C. Pozzolini

¹^,²

and M. Salaun

³

Abstract. Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simpliﬁed model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini’s conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coeﬃcient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated.

Mathematics Subject Classification. 35L85, 65M12, 74H15, 74H45.

Received October 8, 2010.

Published online July 4, 2011.

Introduction

During the launch of a satellite, the level of excitation is very high and the mass reduction of satellite solar arrays can lead to impact problem and possibly damage the structure. So CNES is very interested in the use of Finite Element Method (FEM) including local unilateral contact. The goal is to ensure that numerical simulations are predictive enough to maintain a high reliability of spacecraft structures. This problem has been a subject of intense research over the past twenty years, but the introduced methods are still hard to apply on industrial structures. By the way, in industrial FEM analysis, it is usual to make simplifying assumptions, in the modelling of joints for example, or to neglect some phenomena, such as contact between structures. Then, an updating is introduced to modify some parameters (such as mass, stiﬀness and damping of sub-structures or connections between components) in the numerical model in order to obtain better agreement between numerical and experimental data. To select erroneous parameters, a localization criterion is applied and, classically at

Keywords and phrases.Variational inequalities, ﬁnite element method, elastic beam, dynamics, unilateral constraints, restitution coeﬃcient.

∗ The authors would like to thank Pr. Yves Renard for stimulating discussions. This work has been carried out with the generous support of Centre National d’Etudes Spatiales(Toulouse, France).

1 Pˆole de Math´ematiques, INSA de Lyon, 20 rue Albert Einstein, 69621 Villeurbanne Cedex, France.

cedric.pozzolini@insa-lyon.fr

2 Centre National d’ ´Etudes Spatiales, 18 avenue ´Edouard Belin, 31401 Toulouse, France.

3 Université de Toulouse; INSA, UPS, EMAC, ISAE; ICA (Institut Clément Ader); 10 avenue Édouard Belin, 31055 Toulouse, France. michel.salaun@isae.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2011

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CNES, a constitutive relation error is used. It means that the mechanical system energy is used. In this respect, it is of importance that the numerical simulations does not aﬀect the energy of the system.

As far as elastodynamic contact problems are concerned, it is known that most of usual numerical schemes exhibit spurious oscillations on the contact displacement and stress (see for instance [7]). Moreover, these oscillations do not disappear when the time step decreases. On the contrary, they tend to increase, which makes very difficult to build stable numerical schemes to solve such problems. These difficulties have already led to many researches and a great variety of solutions were proposed. A first idea is to add damping terms but it leads to a loss of accuracy on the solution. However, let us remark that adding damping leads to some existence results, such as [8]. Another way is to implicit the contact stress (see [2,3]) but the kinetic energy of the contacting nodes is lost at each impact. Looking for some energy conserving schemes is now a well-adressed problem, see for example [6,9,10,17]. Nevertheless, these schemes exhibit large oscillations on the contact stress.

Besides, most of them do not strictly respect the constraint. Moreover, the way to establish balance of energy is a mathematicaly very diﬃcult problem even in the “simple” case of viscoelastic barmodel with Signorini condition, see [14,15].

So, the goal of this paper is to introduce energy conserving schemes, based on the singular dynamic method introduced by Renard [16], inspired from [7]. As this method was built for second order problems (Laplace operator or elasticity), we try here to achieve a generalization to fourth order problems such as Euler-Bernouilli beams or Kirchhoﬀ-Love plates. Let us remark that this paper will only adress the case of beams but the case of plates is in progress.

Consequently, this paper will be organized as follows. In the next section, the model problem we adress is described. Then, the so-called singular dynamic method is introduced, for which stable singular mass matrices schemes are derived in the case of our beam model. In Section 3, two full discretized schemes (Midpoint and β-Newmark) are given, with particular emphasis on the way to take into account the restitution coeﬃcient.

Finally, Section 4 presents various numerical results and investigates energy conservation capabilities of the previous schemes.

1. The continuous elastodynamic contact problem

The motion of a beam submitted to an external load is studied. This beam is clamped on its left edge and its vertical displacement is limited by rigid obstacles. Its longitudinal axis, which is its stress free reference conﬁguration, coincides with interval Ω =]0, L[. Euler-Bernouilli model is chosen to represent the motion of the beam under the assumption of small displacements (the relationship between stress and strain is considered to be linear). Then,u(x, t) stands for the vertical displacement of pointxof the beam, at timet. In the following, we will use the notations: u= ^∂u_∂x and ˙u=^∂u_∂t·So the Euler-Bernouilli model for a clamped/free beam reads

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ρS ∂²u

∂t²(x, t) + EI ∂⁴u

∂x⁴(x, t) =f(x, t) ∀(x, t)∈]0, L[×]0, T[ u(x,0) =u₀(x), u(x,˙ 0) =v₀(x) ∀x∈[0, L]

u(0, t) = ∂u

∂x(0, t) = 0 = ∂³u

∂x³(L, t) =∂²u

∂x²(L, t) ∀t∈[0, T]

(1.1)

where ρ (≥ ρ₀ > 0) is the mass density and E is the Young’s modulus of the material, while S and I are respectively the surface and the inertial momentum of the beam section. In the case of a sine-sweep base forced vibration, boundary conditions should be given on the left edge by

u(0, t) =asin(ωt), ∂u

∂x(0, t) = 0, ∀t∈[0, T]. (1.2)

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As we will look for a variational solution of (1.1), we need to introduce some functional spaces. So, we set H=L²(Ω), W={w∈H²(Ω)/ w(0) =w(0) = 0},

whereH²(Ω) is the usual Sobolev space, while the equalitiesw(0) =w(0) = 0 should be understood in the trace sense. Moreover, in the case of dynamic frictionless Euler-Bernouilli model with Signorini contact conditions along the beam, the displacement has to belong to the convex setK ⊂ Wgiven by

K={w∈W/ g₁(x) ≤ w(x) ≤ g₂(x), ∀x∈[0, L]},

whereg₁andg₂ are two mappings from [0, L] to ¯Rsuch that there existsg > 0 such that g₁(x) ≤ −g < 0 < g ≤ g₂(x) ∀x∈[0, L].

Then, the mechanical frictionless elastodynamic problem for a beam between two rigid obstacles can be written as the following variational inequality

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

Find u : [0, T]→Ksuch that for almost everyt∈[0, T] and for everyw∈W

Ω

ρS ∂²u

∂t²(t) (w−u(t)) + EI ∂²u

∂x²(t) ∂²(w−u(t))

∂x²

dΩ ≥

Ω

f (w−u(t)) dΩ u(x,0) =u₀(x)∈K, u(x,˙ 0) =v₀(x), ∀x∈Ω.

(1.3)

Assuming that f ∈ L²(0, T;H), u₀ ∈ K, v₀ ∈ H, Kuttler and Schillor proved in [8] that problem (1.3) has a solution u belonging to L²(0, T;K). For this, they used a penalty method. Another proof of this result is due to Dumont and Paoli [4], who established convergence of the solutions of fully discretized approximations of the problem. As far as uniqueness is concerned, it can be easily shown that it does not occur for (1.3): a counterexample is given in [1]. Finally, there is no result about conservation of energy at the limit.

In fact, discretization of (1.3) does not describe completely the motion: a constitutive law for impact should be added (see [11]). For example, if there is an impact at (x₀, t₀), this law gives the relation between velocities before and after impact

˙

u(x₀, t⁺₀) =−eu(x˙ ₀, t⁻₀) whenever u(x₀, t₀)∈∂K, (1.4) where scalar e, called restitution coefficient, is a real number belonging to [0,1]. e = 1 matches to a perfect impact: velocity is conserved, up to its sign, wherease= 0 is an absorbing shock. Let us remark that, in [13], the authors show that the restitution coefficient for a bar is a rather ill-defined concept. For instance, their numerical experiments underline the observed restitution coefficient depends very strongly on the initial angle of the bar with horizontal. Moreover, in the particular case of a slender bar dropped on a rigid foundation, the chosen value of the restitution coefficient does not seem to have a great influence on the displacement limit when the space step tends to zero.

Despite the previous remarks, our idea is to explicitly incorporate the restitution coeﬃcient into (1.3) and to observe how some numerical schemes will simulate the experimental behavior of a vibrating beam. Doing this, our aim is to investigate the possibility to use restitution coeﬃcient as an unknown parameter for updating of FEM with experimental results.

2. Singular dynamic method 2.1. Well-posed space semi-discretization

The goal of this section is to present a well-posed space semi-discretization of Problem (1.3). As usual, a Galerkin method is used for space discretization, but the original idea, due to Renard [16], is to introduce

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diﬀerent approximations for displacement uand velocityv = ˙u. So, letW^h and H^h be two ﬁnite dimensional vector subspaces of WandHrespectively. Let K^h ⊂ W^h be a closed convex nonempty approximation ofK.

The new approximation of (1.3) reads

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

Find u^h : [0, T]→K^hand v^h : [0, T]→H^hsuch that for allt∈(0, T]

Ω

ρS ∂v^h

∂t (w^h−u^h) +EI ∂²u^h

∂x²

∂²(w^h−u^h)

∂x²

dΩ≥

Ω

f(w^h−u^h)dΩ, ∀w^h∈K^h

ΩρS

v^h−∂u^h

∂t

q^hdΩ = 0, ∀q^h∈H^h u^h(x,0) =u^h₀(x), v^h(x,0) =v₀^h(x), ∀x∈Ω

(2.1)

where u^h₀ ∈ K^h and v₀^h ∈ H^h are approximations of u₀ and v₀ respectively. The case H^h = W^h clearly corresponds to a standard Galerkin approximation of (1.3).

Let us now introduce some basis ofW^h and H^h, say respectivelyφi (1 ≤i≤ NW) and ψi (1 ≤i ≤NH).

The above discrete variational formulation is associated with matrices K (stiﬀness matrix), B and C (mass matrices) of respective sizesN_W² ,NH×NW and N_H², deﬁned by

Kij =

ΩEI φi φj dΩ, Bij =

ΩρS φi ψj dΩ, Cij =

ΩρS ψi ψj dΩ.

The related vectors, sayF,U (componentsui) andV (componentsvi), of sizeNW,NW andNH respectively, are such that

Fi=

Ωf φi dΩ, u^h(t) =

N_W

i=1

ui(t)φi, v^h(t) =

N_H

i=1

vi(t)ψi. Let us remark that the second equation of (2.1) reads

C V(t) =BU˙(t).

Since C is always invertible, we obtain V(t) =C⁻¹ B U˙(t) and, then, ˙V(t) = C⁻¹ B U(t), which allows to¨ eliminateV. So the semi-discretized problem (2.1) is equivalent to

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

FindU : [0, T]→K^h andV : [0, T]→H^h such that for allt∈(0, T] (W−U(t))^T (MU¨(t) +KU(t)) ≥ (W −U(t))^T F, ∀W ∈K^h, CV(t) =B U(t),˙

U(0) =U₀, V(0) =V₀,

(2.2)

whereM is the so-called singular mass matrix deﬁned by

M=B^T C⁻¹ B. (2.3)

Let us now explain how the approximationK^h ofKis obtained. We recall that K={w∈W/ g₁(x) ≤ w(x) ≤ g₂(x), ∀x∈[0, L]}.

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As usual, it is natural to introduce a partition of interval [0, L] intoN subintervals of lengthh=L/N, built on nodesxi =ih, for 0 ≤ i ≤ N. So, unilateral constraints are only considered at these nodes. It means convex K^his

K^h={w^h∈W^h/ g₁(xi) ≤ w^h(xi) ≤ g₂(xi), ∀i∈[0, N]}. (2.4) With vector notations, setting α⁻_i ≡ g₁(xi) andα⁺_i ≡ g₂(xi) for all i, this space may be written (we keep the same notation for simplicity)

K^h={W ∈R^N^W / α⁻_i ≤ (Gⁱ)^T W ≤α⁺_i , ∀i∈[0, N]}, whereGⁱ is the vector ofR^N^W such that (Gⁱ)^T W =w^h(xi), for all nodexi.

Remark 2.1. Since we deal with a fourth order problem with respect to the space derivative, it is not possible to consider a linear space approximation. In fact, for this beam model, we use the classical Hermite third degree polynomials to approximate the numerical displacement. It means the degrees of freedom are node displacements and their derivatives. So, in the above approximation of K, we consider only constraints on node displacements: the eﬀect of the derivatives, namely the curvature, is not taken into account. Then, in this framework, the beam could cross the obstacle between two nodes, but we shall neglect this aspect in the following.

Furthemore, it was told that functionsg₁andg₂takes their values in ¯R, which means thatα^±_i may be equal to ±∞. In this case, the constraint is worthless. For instance, it will be the case if the obstacles are reduced to end stops on the free edge of the beam. Then,α^±_i =±∞for alli = N andα^±_N =±g, if g stands for the allowed maximal displacement. Moreover, it is assumed that the clamped edge, which corresponds to node x₀, satisﬁes the constraints. So, it is natural to introduce the number, sayNG, of “real” constraints. For example, NG= 1 for end stops andNG=N for ﬂat obstacles up and under the beam.

Now, let us denote byGtheNW ×NGmatrix, which components areGij= (Gⁱ)j. As the previous choice is clearly such that vectorsGⁱare linearly independent, using the Lagrange multipliers, the discrete problem (2.2) is also equivalent to the following one

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

Find U : [0, T]→K^h andV : [0, T]→H^h such that for allt∈(0, T] MU(t) +¨ KU(t) =F+

N_G

i=1

λⁱ(t)Gⁱ,

⎧⎪

⎨

⎪⎩

λⁱ(t)≥0, (Gⁱ)^T U(t) ≥ α⁻_i , λⁱ(t)

(Gⁱ)^T U(t)−α⁻_i

= 0 or

λⁱ(t)≤0, (Gⁱ)^T U(t) ≤ α⁺_i , λⁱ(t)

(Gⁱ)^T U(t)−α⁺_i

= 0

⎫⎪

⎬

⎪⎭ 1≤i≤NG

C V(t) =BU˙(t), U(0) =U₀, V(0) =V₀.

(2.5)

Here, the Lagrange multipliersλⁱ are the reaction forces which are measures on (0, T]. And the orthogonality has its natural meaning: an appropriate duality product between the two terms of the relation vanishes.

Now, let us introduce subspaceF^h ofW^h, deﬁned by F^h=

w^h∈W^h

ΩρS w^hξ^hdΩ = 0, ∀ξ^h∈H^h

.

Then, with the above deﬁnitions, we have

F^h= kerB.

The proof of the following result can be found in [16].

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Theorem 2.2. If W^h,H^h andK^h satisfy the following Inf-Sup condition

inf

Q∈R^Ng\{0} sup

W∈F^h\{0}

Q^T GW

Q W > 0, (2.6)

then Problem (2.2)admits a unique solution U(t). Moreover, this solution is Lipschitz-continuous with respect tot and verifies the following persistency condition

λⁱ(t) (Gⁱ)^T U(t) = 0,˙ ∀t∈(0, T], 1≤i≤Ng. Finally, solution U(t)is energy conserving in the sense that the discrete energy

E^h(t) = 1

2 U˙^T(t)M U(t) +˙ 1

2 U^T(t)K U(t) − U^T(t)F, is constant with respect to t.

Remark 2.3. The persistency condition (see [9] and [10]), which links velocity ˙U(t) and Lagrange multipliers, is a stronger condition than the classical complementary condition between solutionU(t) and Lagrange multipliers.

That is this persistency condition which allows to prove energy conservation.

2.2. Numerical discretization

Thanks to the above theorem, proving condition (2.6) is suﬃcient to obtain well-posedness of the discrete problem (2.2). Let us remark this condition is equivalent to the fact that matrixGis surjective on F^h. As a result of this, we must have

dimF^h ≥ NG and consequently dimH^h ≤ dimW^h − NG.

This prescribes conditions on the approximation spaces W^h, H^h and also K^h. In order to illustrate that condition (2.6) holds for interesting practical situations, we will give two examples of approximation spaces for our beam problem.

To build the finite element method, it was introduced a partition of [0, L] into N subintervals of length h= L/N, built on nodes xi = ih, for 0 ≤ i ≤ N. As node x₀ = 0 is clamped, we will omit it from now on and consider that indexivaries between 1 andN. Otherwise, it would introduce small modifications in the following. So, at each nodexi are associated two Hermite piecewise cubic functions, sayφ₂i−1 andφ₂i, defined for 1 ≤ i ≤ N by

φ₂i−1(xj) =δij and φ₂_i₋₁(xj) = 0, φ₂i(xj) = 0 and φ₂_i(xj) =δij,

whereδij is Kronecker symbol. Moreover, functionsφj are chosen of classC¹on [0, L], which insures that each φj belongs to the continuous spaceW. Hence, displacementw^h reads

w^h(x) =

N

i=1

w^h₂_i₋₁φ₂i−1(x) +

N

i=1

w^h₂_iφ₂i(x),

and coeﬃcient w₂^h_i₋₁ gives the value of w^h at node xi while w₂^h_i gives the value of its derivative at the same node. The approximation space for displacements is then

W^h=span{φj, 1≤j≤2N},

which is a subset ofW. In this case, with the previous notations, we haveNW = 2N.

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Now, let us recall that constraints are only considered at nodesxi. Ifw^h is an element ofW^h, we have K^h={w^h∈W^h/ α⁻_i ≤ w^h(xi) =w^h₂_i₋₁ ≤ α⁺_i , ∀i∈[1, N]}.

Let W be the vector ofR^N^W which components arew_j^h. Then, vectorGⁱ ofR^N^W, which components are all zero except (Gⁱ)₂i−1 = 1, is such that (Gⁱ)^T W =w₂^h_i₋₁, for all nodexi. To check the Inf-Sup condition, we will consider the “worst” case, which occurs when the whole beam is between two obstacles. It means that each α^±_i is ﬁnite.

2.2.1. P₀ interpolation for velocity

The ﬁrst choice for spaceH^his to use piecewise constant polynomial functions. Let us begin by characterizing F^h= kerB. Letw^h be an element ofW^h. We have

w^h(x) =

N

i=1

w^h₂_i₋₁φ₂i−1(x) +

N

i=1

w^h₂_iφ₂i(x). (2.7)

w^h belongs toF^h if

ΩρS w^h ξ^hdΩ = 0, ∀ξ^h∈H^h. (2.8)

In the following, we will assume that ρS is constant all along the beam, which allows to drop it. Then, the previous relation is equivalent to x_i

x_i−1

w^h dx= 0, ∀i∈[1, N].

Fori= 1, it becomes x₁

x₀

(w^h₁φ₁(x) + w^h₂φ₂(x)) dx= 0. Asx₀= 0 andx₁=h(mesh size), it is easy to check that φ₂(x) = x²

h²(x−h) and h

0 φ₂(x) dx=−h²

12, which is always non zero. So, it is possible to computew^h₂ fromw^h₁ and we have

w₂^h=−w^h₁ x₁

x₀ φ₁(x) dx x₁

x₀ φ₂(x) dx· Similarly, fori = 1, we have

x_i x_i−1

(w^h₂i−3φ₂i−3(x) + w₂^hi−2φ₂i−2(x) + w^h₂i−1φ₂i−1(x) + w₂^hiφ₂i(x)) dx= 0, which leads to

w^h₂_i =−w^h₂_i₋₃ x_i

x_i−1φ₂i−3(x) dx x_i

x_i−1φ₂i(x) dx − w₂^h_i₋₂ x_i

x_i−1φ₂i(x) dx − w₂^h_i₋₁ x_i

x_i−1φ₂i(x) dx , as x_i

x_i−1φ₂i(x) dx= 0 for the same reason than above. Consequently, asw^h₂ is function ofw₁^h, w^h₄ depends on w^h₁ and w₃^h and, by direct induction, for any element of kerB, the degrees of freedomw^h₂_i can be expressed as functions ofw₂^h_j₋₁. It means any element ¯w^h(x) of kerB reads

¯ w^h(x) =

N

i=1

w₂^hi−1φ₂i−1(x) +

N

i=1

fi(w^h₂j−1)φ₂i(x), wherefi(w^h₂_j₋₁) is a function of the “odd” degrees of freedom.

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Therefore, if ¯W is the vector ofR^N^W which components are ¯w_j^h, it becomes obvious that (Gⁱ)^T W¯ =w₂^h_i₋₁, for all nodexi : Gis surjective on kerBand the Inf-Sup condition is satisﬁed.

Remark 2.4. It is easy to see the previous proof remains true if the mesh sizehis not constant, which means nodesxiare not regularly distributed along the beam. Similarly, the condition “ρSconstant all along the beam”

may be replaced by “ρS constant on each element”.

2.2.2. P₁ interpolation for the velocity

The second choice for space H^h is to use continuous piecewise linear polynomial functions. If N + 1 is the number of nodes on the beam (including the “clamped node”), let us remark that we have in this case dimH^h=N+ 1 which is not less equal than dimW^h − NG= 2N − N =N in the worst case. Nevertheless, the “clamped node” velocity is zero and it is natural to take this condition into account. Then, dimH^h =N and we can hope the Inf-Sup condition may occur.

As above, let us begin by characterizingF^h. Starting from deﬁnitions ofw^h(2.7) andF^h(2.8), and assuming again thatρS is constant which allows to drop it, we obtain

x_i+1 x_i−1

w^hψi dx= 0, ∀i∈[1, N−1], (2.9)

and xN

x_N−1

w^h ψN dx= 0, (2.10)

where ψi is the continuous piecewise linear function such thatψi(xj) =δij. Before analysing these equations, let us ﬁrst give the expressions of the ﬁnite element basis functions, we shall need further. It is made on interval [0, h], the extension to a generic interval [xi, xi+1], where xi =ih, being obtained by a simple translation. So we have for the piecewise linear polynomial functions

ψl(x) = 1 − x

h, ψr(x) =x h,

and for the third order polynomial functions associated with the derivative degrees of freedom φl(x) =x

1 − x

h ₂

, φr(x) = x²

h²(x−h),

where indices landrrepresent the left node and the right one respectively. Finally, it is easy to check that

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ h

0 φl(x)ψl(x) dx=h² 20 =−

h

0 φr(x)ψr(x) dx, h

0 φl(x)ψr(x) dx=h² 30 =−

h

0 φr(x)ψl(x) dx.

(2.11)

Then, fori= 1, by using (2.11), equation (2.9) becomes x₂

x₀

w^h ψ₁ dx = x₂

x₀

(w^h₁φ₁ + w₂^hφ₂)ψ₁ dx + x₂

x₁

(w₃^hφ₃ + w^h₄φ₄)ψ₁ dx

= w^h₁ x₂

x₀

φ₁ ψ₁ dx + w^h₂

−h² 20 + h²

20

+ w₃^h x₂

x₁

φ₃ ψ₁ dx − w₄^hh² 30

= 0.

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So, it is possible to compute w^h₄ from w₁^h and w₃^h as w₄^h =f(w^h₁, w^h₃). For simplicity, in the following,f will indicate various (polynomial) functions. Similarly, for 1 < i < N, we have

x_i+1 x_i−1

w^hψi dx = xi

x_i−1

(w^h₂_i₋₃φ₂i−3+w^h₂_i₋₂φ₂i−2)ψi dx+ x_i+1

x_i−1

(w₂^h_i₋₁φ₂i−1+w^h₂_iφ₂i)ψi dx +

x_i+1 x_i

(w^h₂i+1φ₂i+1+w^h₂i+2φ₂i+2)ψi dx

= w₂^h_i₋₃ x_i

x_i−1

φ₂i−3 ψi dx+w^h₂_i₋₂ h²

30+w^h₂_i₋₁ x_i+1

x_i−1

φ₂i−1 ψi dx + w₂^hi

−h² 20 + h²

20

+w₂^hi+1

x_i+1 x_i

φ₂i+1 ψi dx−w^h₂i+2

h² 30

= 0

which leads to

w^h₂_i₊₂=w^h₂_i₋₂ + f(w^h₂_i₋₃, w^h₂_i₋₁, w^h₂_i₊₁).

Consequently, by direct induction, we obtain

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

w^h₂ = w^h₂

w^h₄ = f(w₁^h, w^h₃)

w^h₆ = w^h₂ + f(w₁^h, w^h₃, w₅^h)

w^h₈ = w^h₄ + f(w₃^h, w^h₅, w₇^h) =f(w^h₁, w^h₃, w^h₅, w^h₇)

w^h₁₀ = w^h₆ + f(w₅^h, w^h₇, w₉^h) =w^h₂ + f(w^h₁, w₃^h, w^h₅, w^h₇, w^h₉) ...

w^h₄_k₋₂ = w^h₂ + f(w_odd^h ) w^h₄_k = f(w_odd^h )

...

wheref(w^h_odd) stands for various functions depending on the set ofw_j^h,jbeing odd. In particular, let us observe that the two last equations, corresponding toi=N − 1, read

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

w^h₂_N₋₂ = w₂^h + f(w_odd^h ) w^h₂_N = f(w_odd^h )

if N is even w^h₂_N₋₂ = f(w^h_odd)

w₂^h_N = w^h₂ + f(w^h_odd)

if N is odd.

(2.12)

Finally, the last equation (2.10) becomes x_N

x_N−1

w^h ψN dx = x_N

x_N−1

(w^h₂N−3φ₂N−3+w^h₂N−2φ₂N−2+w₂^hN−1φ₂N−1+w₂^hNφ₂N)ψN dx

= w^h₂_N₋₃ xN

x_N−1

φ₂N−3ψN dx+w₂^h_N₋₂ h²

30+w^h₂_N₋₁ x_N+1

x_N−1

φ₂N−1 ψN dx

− w^h₂N

h²

= 0, 20 or else

w^h₂N = 2

3w^h₂N−2 + f(w^h₂N−3, w^h₂N−1).

Then, using (2.12), in both cases (N being odd or even), we obtainw^h₂ =f(w^h_odd).

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Finally, exactly as in Section2.2.1, for any element of kerB, the degrees of freedomw₂^h_i can be expressed as functions ofw₂^h_j₋₁ and the Inf-Sup condition is satisﬁed.

3. Full discretized schemes

In this section, we present two approaches for space discretization of the velocity, and we compare Midpoint andβ-Newmark schemes for time discretization. These schemes are interesting since they are energy conserving during the linear part of the motion (equation without constraint). As previously mentioned, classical P₃- Hermite ﬁnite elements are used to approximate displacementu. Moreover, Δt will be the time step andethe restitution coeﬃcient.

3.1. Newmark-Dumont-Paoli schemes

3.1.1. Case of a regular mass matrix

To solve problem (1.3), Dumont-Paoli [4] introduced the following fully implicit Newmark scheme

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

Find Uⁿ⁺¹∈K^h such that for allW ∈K^h W −Uⁿ⁺¹T

Mr

Uⁿ⁺¹−2Uⁿ+Uⁿ⁻¹

Δt² + K

βUⁿ⁺¹+ (1−2β)Uⁿ+βUⁿ⁻¹

≥

W −Uⁿ⁺¹T

F^n,β

(3.1)

where

F^n,β =βFⁿ⁺¹+ (1−2β)Fⁿ+βFⁿ⁻¹, (3.2) F^k being the vector which components are F_i^k =

Ωf(x, kΔt)φi(x) dΩ. Finally, (φi)_i stand for the piecewise cubic basis functions deﬁning spaceW^handMris the associated regular mass matrix.

To take into account the restitution coefficient e defined by (1.4), we follow the choice introduced by Paoli-Schatzman (see [12,13]), which consists in replacing Uⁿ⁺¹ by Ûⁿ⁺¹₁₊⁺êU_e ⁿ⁻¹· Then, a more general discretization of (1.3) becomes

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

FindUⁿ⁺¹^,e ≡ Uⁿ⁺¹+eUⁿ⁻¹

1 +e ∈K^hsuch that for allW ∈K^h (W−Uⁿ⁺¹^,e)^T

Mr

Uⁿ⁺¹−2Uⁿ+Uⁿ⁻¹

Δt² +K

βUⁿ⁺¹+ (1−2β)Uⁿ+βUⁿ⁻¹

≥

W−Uⁿ⁺¹^,eT

F^n,β.

(3.3)

Let us remark that Dumont-Paoli fully implicit scheme corresponds to e = 0, which is a totally absorbing impact. In this case, the authors established unconditional stability forβ = 1/2 whereas a conditional stability result is obtained whenβ ∈ [0,1/2[ (see [4]). Moreover, forβ ∈ [0,1/2], a weak convergence result (up to a subsequence) is demonstrated.

Remark 3.1. Deﬁning the total energy by E(w,w) :=˙

Ω

ρS

2 ( ˙w(x, t))² + EI

2 (w(x, t))² − f(x, t)w(x, t)

dΩ,

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it is easy to see that E(uⁿ⁺¹, vⁿ⁺¹) ≤ E(uⁿ, vⁿ). This Newmark-Dumont-Paoli scheme is dissipative in energy. Indeed, energy is conserved as long as the beam does not touch the obstacles and it is dissipated when beam reaches them, except for a totally elastic shock (e = 1) which is the only case where kinetic energy is conserved.

3.1.2. Case of singular mass matrix

To derive a Newmark scheme using the singular mass matrix approach, let us go from the equilibrium equation given in (2.5)

MU¨ + K U =F + Λ ≡ F ,˜

where Λ stands for the reaction, which is zero when there is no contact. Moreover, the singular mass approach introduces matricesC andBsuch thatC V =BU˙, the singular mass matrix beingM=B^T C⁻¹ B.

The usual (1/2, β)-Newmark scheme reads

⎧⎨

⎩

Uⁿ⁺¹ = Uⁿ + ΔtU˙ⁿ + (¹₂−β) Δt²U¨ⁿ + βΔt²U¨ⁿ⁺¹, U˙ⁿ⁺¹ = U˙ⁿ + ^Δ₂^tU¨ⁿ + ^Δ₂^tU¨ⁿ⁺¹.

Multiplying left by Band usingCV =BU˙, we deduce

⎧⎨

⎩

BUⁿ⁺¹ = BUⁿ + ΔtCVⁿ + (¹₂−β) Δt²BU¨ⁿ + βΔt²BU¨ⁿ⁺¹, CVⁿ⁺¹ = CVⁿ + ^Δ₂^tB U¨ⁿ + ^Δ₂^tBU¨ⁿ⁺¹.

(3.4)

It is well-known it is possible to derive a two-step scheme by eliminating velocity from this relations. First, we

write

BUⁿ = B Uⁿ⁻¹ + Δt CVⁿ⁻¹ + (¹₂−β) Δt²BU¨ⁿ⁻¹ + βΔt²BU¨ⁿ, BUⁿ⁺¹ = B Uⁿ + Δt CVⁿ + (¹₂−β) Δt²BU¨ⁿ + βΔt²BU¨ⁿ⁺¹, which leads to

B(Uⁿ⁺¹ − 2Uⁿ + Uⁿ⁻¹) = ΔtC (Vⁿ − Vⁿ⁻¹) + Δt²B

βU¨ⁿ⁺¹ + ₁

2−2βU¨ⁿ − ₁

2−βU¨ⁿ⁻¹

, and ﬁnally, with the second equation of (3.4), written at stepninstead ofn+ 1

B(Uⁿ⁺¹ − 2Uⁿ + Uⁿ⁻¹) = Δt² B

βU¨ⁿ⁺¹ + (1−2β) ¨Uⁿ + βU¨ⁿ⁻¹

. Multiplying this relation byB^T C⁻¹, we obtain

M(Uⁿ⁺¹ − 2Uⁿ + Uⁿ⁻¹) = Δt² M

βU¨ⁿ⁺¹ + (1−2β) ¨Uⁿ + βU¨ⁿ⁻¹

,

whereMis the singular mass matrix. As usual for Newmark scheme, we replace acceleration by its value, given by the equilibrium equation. Hence, in the case of the singular mass matrix approach, Newmark scheme reads

M Uⁿ⁺¹ − 2 Uⁿ + Uⁿ⁻¹

Δt² +K

βUⁿ⁺¹ + (1−2β)Uⁿ + βUⁿ⁻¹

=

βF˜ⁿ⁺¹ + (1−2β) ˜Fⁿ + βF˜ⁿ⁻¹

, which reads exactly as the Newmark scheme except the regular mass matrix Mr has been replaced by the singular one.